Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480319 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| socle | C2 | C2×#1799 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×M16 ◁ C2×#568 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1130 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,11)(4,15)(5,18,19,8)(6,10)(7,13)(9,17)(12,24)(14,28,26,23)(16,22,30,29)(20,27,31,32)(21,25);(1,3,4,13,6,12,17,25)(2,7,9,11,10,21,15,24)(5,14,16,27,19,26,30,32)(8,20,22,23,18,31,29,28);(2,8)(7,20)(9,22)(10,18)(11,23)(15,29)(21,31)(24,28);(3,12)(7,21)(11,24)(13,25)(14,26)(20,31)(23,28)(27,32)Polycyclic presentation:
2^9|a^2=g,b^2=e,e^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,d^b=di,g^c=gh,g^f=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72236890, #72237535, #72237542, … |
| maximal normal subgroup of | #72236890, #72237535, #72237542, … |
| maximal quotient of | #72303813, C2×#493733, #72312577 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400929);